Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 1,648 | 1,652 | theorem extendWith_zero [DecidableEq ι] [∀ i, Zero (α i)] (x : α none) :
(0 : Π₀ i, α (some i)).extendWith x = single none x := by |
ext (_ | j)
· rw [extendWith_none, single_eq_same]
· rw [extendWith_some, single_eq_of_ne (Option.some_ne_none _).symm, zero_apply]
|
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 820 | 825 | theorem transferTransversal_apply' (q : orbitRel.Quotient (zpowers g) (G ⧸ H))
(k : ZMod (minimalPeriod (g • ·) q.out')) :
↑(toEquiv (transferTransversal H g).2 (g ^ (cast k : ℤ) • q.out')) =
g ^ (cast k : ℤ) * q.out'.out' := by |
rw [transferTransversal_apply, transferFunction_apply, ← quotientEquivSigmaZMod_symm_apply,
apply_symm_apply]
|
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 358 | 369 | theorem toList_formPerm_nontrivial (l : List α) (hl : 2 ≤ l.length) (hn : Nodup l) :
toList (formPerm l) (l.get ⟨0, (zero_lt_two.trans_le hl)⟩) = l := by |
have hc : l.formPerm.IsCycle := List.isCycle_formPerm hn hl
have hs : l.formPerm.support = l.toFinset := by
refine support_formPerm_of_nodup _ hn ?_
rintro _ rfl
simp [Nat.succ_le_succ_iff] at hl
rw [toList, hc.cycleOf_eq (mem_support.mp _), hs, card_toFinset, dedup_eq_self.mpr hn]
· refine ext_get... |
import Mathlib.Analysis.Convex.StrictConvexSpace
#align_import analysis.convex.uniform from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open Set Metric
open Convex Pointwise
class UniformConvexSpace (E : Type*) [SeminormedAddCommGroup E] : Prop where
uniform_convex : ∀ ⦃ε : ℝ⦄,
... | Mathlib/Analysis/Convex/Uniform.lean | 60 | 112 | theorem exists_forall_closed_ball_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y⦄, ‖y‖ ≤ 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ := by |
have hε' : 0 < ε / 3 := div_pos hε zero_lt_three
obtain ⟨δ, hδ, h⟩ := exists_forall_sphere_dist_add_le_two_sub E hε'
set δ' := min (1 / 2) (min (ε / 3) <| δ / 3)
refine ⟨δ', lt_min one_half_pos <| lt_min hε' (div_pos hδ zero_lt_three), fun x hx y hy hxy => ?_⟩
obtain hx' | hx' := le_or_lt ‖x‖ (1 - δ')
· rw... |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 93 | 98 | theorem funMulInvSnorm_rpow {p : ℝ} (hp0 : 0 < p) {f : α → ℝ≥0∞} {a : α} :
funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ c, f c ^ p ∂μ)⁻¹ := by |
rw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)]
suffices h_inv_rpow : ((∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ c : α, f c ^ p ∂μ)⁻¹ by
rw [h_inv_rpow]
rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one]
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
namespace CliffordAlgebra
variable {R M : Type*} [Co... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 207 | 218 | theorem even_induction {motive : ∀ x, x ∈ evenOdd Q 0 → Prop}
(algebraMap : ∀ r : R, motive (algebraMap _ _ r) (SetLike.algebraMap_mem_graded _ _))
(add : ∀ x y hx hy, motive x hx → motive y hy → motive (x + y) (Submodule.add_mem _ hx hy))
(ι_mul_ι_mul :
∀ m₁ m₂ x hx,
motive x hx →
m... |
refine evenOdd_induction (motive := motive) (fun rx => ?_) add ι_mul_ι_mul x hx
rintro ⟨r, rfl⟩
exact algebraMap r
|
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 99 | 115 | theorem Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
refine Subtype.coe_injective (res_injective ?_)
dsimp
ext n : 1
induction' n with n ih; · simp
simp only [res_succ, cons.injEq]
refine ⟨?_, ih⟩
contrapose hA
simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
refine ⟨res x n, _, _, hA, ?_⟩
r... |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
assert_not_exists HasFDerivAt
assert_not_exists ConformalAt
noncom... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 297 | 305 | theorem norm_sub_eq_add_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
‖x - y‖ = ‖x‖ + ‖y‖ ↔ angle x y = π := by |
refine ⟨fun h => ?_, norm_sub_eq_add_norm_of_angle_eq_pi⟩
rw [← inner_eq_neg_mul_norm_iff_angle_eq_pi hx hy]
obtain ⟨hxy₁, hxy₂⟩ := norm_nonneg (x - y), add_nonneg (norm_nonneg x) (norm_nonneg y)
rw [← sq_eq_sq hxy₁ hxy₂, norm_sub_pow_two_real] at h
calc
⟪x, y⟫ = (‖x‖ ^ 2 + ‖y‖ ^ 2 - (‖x‖ + ‖y‖) ^ 2) / 2... |
import Mathlib.Algebra.Order.Group.PosPart
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Order.Lattice
#align_import analysis.normed.order.lattice from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
-- Porting note: this now exists as a global notation
-- local n... | Mathlib/Analysis/Normed/Order/Lattice.lean | 96 | 103 | theorem dual_solid (a b : α) (h : b ⊓ -b ≤ a ⊓ -a) : ‖a‖ ≤ ‖b‖ := by |
apply solid
rw [abs]
nth_rw 1 [← neg_neg a]
rw [← neg_inf]
rw [abs]
nth_rw 1 [← neg_neg b]
rwa [← neg_inf, neg_le_neg_iff, inf_comm _ b, inf_comm _ a]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 247 | 252 | theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
∫⁻ _x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [lintegral_const, laverage_eq,
ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,489 | 1,492 | theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
interior (pi I s) = I.pi fun i => interior (s i) := by |
ext a
simp only [Set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI]
|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 489 | 490 | theorem diagonal_subset_iff {s} : diagonal α ⊆ s ↔ ∀ x, (x, x) ∈ s := by |
rw [← range_diag, range_subset_iff]
|
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 184 | 185 | theorem measure_symmDiff_eq_zero_iff {s t : Set α} : μ (s ∆ t) = 0 ↔ s =ᵐ[μ] t := by |
simp [ae_eq_set, symmDiff_def]
|
import Mathlib.Data.Finset.Order
import Mathlib.Algebra.DirectSum.Module
import Mathlib.RingTheory.FreeCommRing
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.SuppressCompilation
#align_import algebra.direct_limit from "leanprover-community/mathlib"@"f0c8bf9245297a... | Mathlib/Algebra/DirectLimit.lean | 664 | 689 | theorem of.zero_exact_aux2 {x : FreeCommRing (Σi, G i)} {s t} [DecidablePred (· ∈ s)]
[DecidablePred (· ∈ t)] (hxs : IsSupported x s) {j k} (hj : ∀ z : Σi, G i, z ∈ s → z.1 ≤ j)
(hk : ∀ z : Σi, G i, z ∈ t → z.1 ≤ k) (hjk : j ≤ k) (hst : s ⊆ t) :
f' j k hjk (lift (fun ix : s => f' ix.1.1 j (hj ix ix.2) ix.1.... |
refine Subring.InClosure.recOn hxs ?_ ?_ ?_ ?_
· rw [(restriction _).map_one, (FreeCommRing.lift _).map_one, (f' j k hjk).map_one,
(restriction _).map_one, (FreeCommRing.lift _).map_one]
· -- Porting note: Lean 3 had `(FreeCommRing.lift _).map_neg` but I needed to replace it with
-- `RingHom.map_neg` to ... |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 139 | 153 | theorem monomial_smul_apply (i : ℕ) (r : R) (g : PolynomialModule R M) (n : ℕ) :
(monomial i r • g) n = ite (i ≤ n) (r • g (n - i)) 0 := by |
induction' g using PolynomialModule.induction_linear with p q hp hq
· simp only [smul_zero, zero_apply, ite_self]
· simp only [smul_add, add_apply, hp, hq]
split_ifs
exacts [rfl, zero_add 0]
· rw [monomial_smul_single, single_apply, single_apply, smul_ite, smul_zero, ← ite_and]
congr
rw [eq_iff... |
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 592 | 602 | theorem comp_C_integral_of_surjective_of_jacobson {S : Type*} [Field S] (f : R[X] →+* S)
(hf : Function.Surjective ↑f) : (f.comp C).IsIntegral := by |
haveI : f.ker.IsMaximal := RingHom.ker_isMaximal_of_surjective f hf
let g : R[X] ⧸ (RingHom.ker f) →+* S := Ideal.Quotient.lift (RingHom.ker f) f fun _ h => h
have hfg : g.comp (Quotient.mk (RingHom.ker f)) = f := ringHom_ext' rfl rfl
rw [← hfg, RingHom.comp_assoc]
refine (quotient_mk_comp_C_isIntegral_of_ja... |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 226 | 228 | theorem vars_monomial_single (i : σ) {e : ℕ} {r : R} (he : e ≠ 0) (hr : r ≠ 0) :
(monomial (Finsupp.single i e) r).vars = {i} := by |
rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]
|
import Mathlib.FieldTheory.Perfect
#align_import field_theory.perfect_closure from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
open Function
section
variable (K : Type u) [CommRing K] (p : ℕ) [Fact p.Prime] [CharP K p]
@[mk_iff]
inductive PerfectClosure.R : ℕ × K → ℕ... | Mathlib/FieldTheory/PerfectClosure.lean | 328 | 350 | theorem mk_eq_iff (x y : ℕ × K) :
mk K p x = mk K p y ↔ ∃ z, (frobenius K p)^[y.1 + z] x.2 = (frobenius K p)^[x.1 + z] y.2 := by |
constructor
· intro H
replace H := Quot.exact _ H
induction H with
| rel x y H => cases' H with n x; exact ⟨0, rfl⟩
| refl H => exact ⟨0, rfl⟩
| symm x y H ih => cases' ih with w ih; exact ⟨w, ih.symm⟩
| trans x y z H1 H2 ih1 ih2 =>
cases' ih1 with z1 ih1
cases' ih2 with z2 ih2
... |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 128 | 129 | theorem Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ := by |
rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]
|
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
#align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
open Function
open Function (Commute)
namespace Function
section Preorder
variable {α : Type*} [Preorder α] {f : α → α}
| Mathlib/Order/Iterate.lean | 134 | 135 | theorem id_le_iterate_of_id_le (h : id ≤ f) (n : ℕ) : id ≤ f^[n] := by |
simpa only [iterate_id] using monotone_id.iterate_le_of_le h n
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 868 | 872 | theorem finprod_mem_insert' (f : α → M) (h : a ∉ s) (hs : (s ∩ mulSupport f).Finite) :
∏ᶠ i ∈ insert a s, f i = f a * ∏ᶠ i ∈ s, f i := by |
rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton]
· rwa [disjoint_singleton_left]
· exact (finite_singleton a).inter_of_left _
|
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 118 | 121 | theorem Path.cast_eq_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v)
(p' : Path u' v') : p.cast hu hv = p' ↔ HEq p p' := by |
rw [Path.cast_eq_cast]
exact _root_.cast_eq_iff_heq
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 91 | 93 | theorem dom_inv : r.inv.dom = r.codom := by |
ext x
rfl
|
import Mathlib.RingTheory.Coalgebra.Hom
import Mathlib.RingTheory.Bialgebra.Basic
open TensorProduct Bialgebra
universe u v w
structure BialgHom (R A B : Type*) [CommSemiring R]
[Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
[CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗc[R] B, A →* B
... | Mathlib/RingTheory/Bialgebra/Hom.lean | 145 | 147 | theorem coe_coalgHom_mk {f : A →ₗc[R] B} (h h₁) :
((⟨f, h, h₁⟩ : A →ₐc[R] B) : A →ₗc[R] B) = f := by |
rfl
|
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 267 | 270 | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by |
classical
induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 1,095 | 1,099 | theorem tendsto_toReal_iff {ι} {fi : Filter ι} {f : ι → ℝ≥0∞} (hf : ∀ i, f i ≠ ∞) {x : ℝ≥0∞}
(hx : x ≠ ∞) : Tendsto (fun n => (f n).toReal) fi (𝓝 x.toReal) ↔ Tendsto f fi (𝓝 x) := by |
lift f to ι → ℝ≥0 using hf
lift x to ℝ≥0 using hx
simp [tendsto_coe]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 1,637 | 1,640 | theorem map_surjective_iff {f : α → β} : Surjective (map f) ↔ Surjective f := by |
refine ⟨fun h x => ?_, (·.list_map)⟩
let ⟨[y], hxy⟩ := h [x]
exact ⟨_, List.singleton_injective hxy⟩
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 922 | 925 | theorem ncard_le_ncard_iff_ncard_diff_le_ncard_diff (hs : s.Finite := by | toFinite_tac)
(ht : t.Finite := by toFinite_tac) : s.ncard ≤ t.ncard ↔ (s \ t).ncard ≤ (t \ s).ncard := by
rw [← ncard_inter_add_ncard_diff_eq_ncard s t hs, ← ncard_inter_add_ncard_diff_eq_ncard t s ht,
inter_comm, add_le_add_iff_left]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 321 | 323 | theorem dual_node3R (l : Ordnode α) (x : α) (m : Ordnode α) (y : α) (r : Ordnode α) :
dual (node3R l x m y r) = node3L (dual r) y (dual m) x (dual l) := by |
simp [node3L, node3R, dual_node', add_comm]
|
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.TensorProduct.Basic
section Modules
open TensorProduct LinearMap
section Algebras
open Algebra.TensorProduct
open scoped TensorProduct
variable
{R : Type*} [CommSemiring R]
{A B : Type*} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
lemm... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 680 | 699 | theorem Algebra.TensorProduct.map_ker (hf : Function.Surjective f) (hg : Function.Surjective g) :
RingHom.ker (map f g) =
(RingHom.ker f).map (Algebra.TensorProduct.includeLeft : A →ₐ[R] A ⊗[R] C) ⊔
(RingHom.ker g).map (Algebra.TensorProduct.includeRight : C →ₐ[R] A ⊗[R] C) := by |
-- rewrite map f g as the composition of two maps
have : map f g = (map f (AlgHom.id R D)).comp (map (AlgHom.id R A) g) := ext rfl rfl
rw [this]
-- this needs some rewriting to RingHom
simp only [AlgHom.coe_ker, AlgHom.comp_toRingHom]
rw [← RingHom.comap_ker]
simp only [← AlgHom.coe_ker]
-- apply one s... |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 51 | 55 | theorem norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)) := by |
rw [Complex.norm_eq_abs, Complex.abs_exp, neg_mul, neg_re, ← re_add_im b]
simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im]
ring_nf
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 213 | 213 | theorem bernoulli_one : bernoulli 1 = -1 / 2 := by | norm_num [bernoulli]
|
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set.Countable
import Mathlib.Logic.Small.Set
import Mathlib.Order.SuccPred.CompleteLinearOrder
import Mathlib.SetTheory.Cardinal.SchroederBernstein
#align_import set_theory.cardinal.basic f... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,254 | 2,261 | theorem mk_eq_two_iff' (x : α) : #α = 2 ↔ ∃! y, y ≠ x := by |
rw [mk_eq_two_iff]; constructor
· rintro ⟨a, b, hne, h⟩
simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
rcases h x with (rfl | rfl)
exacts [⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
· rintro ⟨y, hne, hy⟩
exact ⟨x, y, hne.symm, eq_un... |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 104 | 108 | theorem hσ'_eq_zero {q n m : ℕ} (hnq : n < q) (hnm : c.Rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m]) = 0 := by |
simp only [hσ', hσ]
split_ifs
exact zero_comp
|
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Linear.Basic
#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
variable (R : Type*) [Semiring R]
class Functor.Linear... | Mathlib/CategoryTheory/Linear/LinearFunctor.lean | 53 | 54 | theorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by |
apply map_smul
|
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 1,496 | 1,498 | theorem isGLB_biInf {s : Set β} {f : β → α} : IsGLB (f '' s) (⨅ x ∈ s, f x) := by |
simpa only [range_comp, Subtype.range_coe, iInf_subtype'] using
@isGLB_iInf α s _ (f ∘ fun x => (x : β))
|
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 178 | 178 | theorem tail_suffix (l : List α) : tail l <:+ l := by | rw [← drop_one]; apply drop_suffix
|
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 64 | 72 | theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
(hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∃ r : ℝ, ∀ᶠ y in 𝓝 x,
r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } := by |
have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
(eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry
rcases this.exists_gt with ⟨r, hr0, hr⟩
refine ⟨r, hr.mono fun y hy => ⟨hr0, ?_⟩⟩
rwa [mem_preimage, mem_iInter₂]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 337 | 341 | theorem oangle_smul_smul_self_of_nonneg (x : V) {r₁ r₂ : ℝ} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
o.oangle (r₁ • x) (r₂ • x) = 0 := by |
rcases hr₁.lt_or_eq with (h | h)
· simp [h, hr₂]
· simp [h.symm]
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 205 | 211 | theorem Ico_image_const_sub_eq_Ico (hac : a ≤ c) :
((Ico a b).image fun x => c - x) = Ico (c + 1 - b) (c + 1 - a) := by |
ext x
simp_rw [mem_image, mem_Ico]
refine ⟨?_, fun h ↦ ⟨c - x, by omega⟩⟩
rintro ⟨x, hx, rfl⟩
omega
|
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {α : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 604 | 613 | theorem lowerSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x := by |
classical
induction' a using Finset.induction_on with i a ia IH
· exact lowerSemicontinuousWithinAt_const
· simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self i a))
(IH fun j ja => ha j (Finset.mem_insert_of_mem ja)... |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 509 | 510 | theorem _root_.Wbtw.oangle₃₁₂_eq_zero {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) : ∡ p₃ p₁ p₂ = 0 := by |
rw [oangle_eq_zero_iff_oangle_rev_eq_zero, h.oangle₂₁₃_eq_zero]
|
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,178 | 1,185 | theorem contDiffOn_succ_iff_hasFDerivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x := by |
rw [contDiffOn_succ_iff_fderivWithin hs]
refine ⟨fun h => ⟨fderivWithin 𝕜 f s, h.2, fun x hx => (h.1 x hx).hasFDerivWithinAt⟩, fun h => ?_⟩
rcases h with ⟨f', h1, h2⟩
refine ⟨fun x hx => (h2 x hx).differentiableWithinAt, fun x hx => ?_⟩
exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) h... |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7e8fc6b22f0d6a"
open Function
open UniformConvergence
@[to_additive]
| Mathlib/Topology/Algebra/Equicontinuity.lean | 20 | 31 | theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
[UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
(hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
Equicontinuous ((↑) ∘ F) := by |
rw [equicontinuous_iff_continuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact continuous_of_continuousAt_on... |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
#align_import combinatorics.simple_graph.clique from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Finset Fintype Function SimpleGraph.Walk... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 173 | 174 | theorem isNClique_one : G.IsNClique 1 s ↔ ∃ a, s = {a} := by |
simp only [isNClique_iff, card_eq_one, and_iff_right_iff_imp]; rintro ⟨a, rfl⟩; simp
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 110 | 117 | theorem differentiableWithinAt_iff_restrictScalars (hf : DifferentiableWithinAt 𝕜 f s x)
(hs : UniqueDiffWithinAt 𝕜 s x) : DifferentiableWithinAt 𝕜' f s x ↔
∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderivWithin 𝕜 f s x := by |
constructor
· rintro ⟨g', hg'⟩
exact ⟨g', hs.eq (hg'.restrictScalars 𝕜) hf.hasFDerivWithinAt⟩
· rintro ⟨f', hf'⟩
exact ⟨f', hf.hasFDerivWithinAt.of_restrictScalars 𝕜 hf'⟩
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 897 | 926 | theorem oangle_sign_smul_add_right (x y : V) (r : ℝ) :
(o.oangle x (r • x + y)).sign = (o.oangle x y).sign := by |
by_cases h : o.oangle x y = 0 ∨ o.oangle x y = π
· rwa [Real.Angle.sign_eq_zero_iff.2 h, Real.Angle.sign_eq_zero_iff,
oangle_smul_add_right_eq_zero_or_eq_pi_iff]
have h' : ∀ r' : ℝ, o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ π := by
intro r'
rwa [← o.oangle_smul_add_right_eq_zero_or_eq... |
import Mathlib.FieldTheory.Tower
import Mathlib.RingTheory.Algebraic
import Mathlib.FieldTheory.Minpoly.Basic
#align_import field_theory.intermediate_field from "leanprover-community/mathlib"@"c596622fccd6e0321979d94931c964054dea2d26"
open FiniteDimensional Polynomial
open Polynomial
variable (K L L' : Type*) [... | Mathlib/FieldTheory/IntermediateField.lean | 344 | 348 | theorem coe_sum {ι : Type*} [Fintype ι] (f : ι → S) : (↑(∑ i, f i) : L) = ∑ i, (f i : L) := by |
classical
induction' (Finset.univ : Finset ι) using Finset.induction_on with i s hi H
· simp
· rw [Finset.sum_insert hi, AddMemClass.coe_add, H, Finset.sum_insert hi]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 682 | 685 | theorem restrict_apply {i : Set α} (hi : MeasurableSet i) {j : Set α} (hj : MeasurableSet j) :
v.restrict i j = v (j ∩ i) := by |
rw [restrict, dif_pos hi]
exact if_pos hj
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 348 | 349 | theorem mk'_eq_zero_iff (x : R) (s : M) : mk' S x s = 0 ↔ ∃ m : M, ↑m * x = 0 := by |
rw [← (map_units S s).mul_left_inj, mk'_spec, zero_mul, map_eq_zero_iff M]
|
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat → Nat
rank_lt : ∀ i, (parent i).1 ≠ i → rank i < rank (parent i)
structure UFNode (α : Type*) where
parent : Nat
value : α
rank : Nat
inductive... | Mathlib/Data/UnionFind.lean | 73 | 77 | theorem mk' {arr : Array α} {f : α → β} {n} {g : Fin n → β} (e : n = arr.size)
(H : ∀ i h₁ h₂, f (arr.get ⟨i, h₁⟩) = g ⟨i, h₂⟩) : Agrees arr f g := by |
cases e
have : (fun i ↦ f (arr.get i)) = g := by funext ⟨i, h⟩; apply H
cases this; constructor
|
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 128 | 134 | theorem isUnit_res_toΓSpecMapBasicOpen : IsUnit (X.toToΓSpecMapBasicOpen r r) := by |
convert
(X.presheaf.map <| (eqToHom <| X.toΓSpecMapBasicOpen_eq r).op).isUnit_map
(X.toRingedSpace.isUnit_res_basicOpen r)
-- Porting note: `rw [comp_apply]` to `erw [comp_apply]`
erw [← comp_apply, ← Functor.map_comp]
congr
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 187 | 190 | theorem restrict_restrict_of_subset (h : s ⊆ t) : (μ.restrict t).restrict s = μ.restrict s := by |
ext1 u hu
rw [restrict_apply hu, restrict_apply hu, restrict_eq_self]
exact inter_subset_right.trans h
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,080 | 1,085 | theorem tendsto_cos_pi_div_two : Tendsto cos (𝓝[<] (π / 2)) (𝓝[>] 0) := by |
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
· convert continuous_cos.continuousWithinAt.tendsto
simp
· filter_upwards [Ioo_mem_nhdsWithin_Iio
(right_mem_Ioc.mpr (neg_lt_self pi_div_two_pos))] with x hx using cos_pos_of_mem_Ioo hx
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variab... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 754 | 757 | theorem one_lt_rpow_iff (hx : 0 ≤ x) : 1 < x ^ y ↔ 1 < x ∧ 0 < y ∨ 0 < x ∧ x < 1 ∧ y < 0 := by |
rcases hx.eq_or_lt with (rfl | hx)
· rcases _root_.em (y = 0) with (rfl | hy) <;> simp [*, lt_irrefl, (zero_lt_one' ℝ).not_lt]
· simp [one_lt_rpow_iff_of_pos hx, hx]
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 101 | 103 | theorem frontier_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (closedBall x r) = sphere x r := by |
rw [frontier, closure_closedBall, interior_closedBall x hr, closedBall_diff_ball]
|
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 529 | 533 | theorem eq_withDensity_rnDeriv₀ {s : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f ν) (hs : s ⟂ₘ ν) (hadd : μ = s + ν.withDensity f) :
ν.withDensity f = ν.withDensity (μ.rnDeriv ν) := by |
rw [withDensity_congr_ae hf.ae_eq_mk] at hadd ⊢
exact eq_withDensity_rnDeriv hf.measurable_mk hs hadd
|
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 198 | 202 | theorem add_cases : Irrational (x + y) → Irrational x ∨ Irrational y := by |
delta Irrational
contrapose!
rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩
exact ⟨rx + ry, cast_add rx ry⟩
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 437 | 438 | theorem id_eq_nilFun {α : TypeVec 0} : @id _ α = nilFun := by |
ext x; cases x
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 851 | 853 | theorem floor_eq_iff : ⌊a⌋ = z ↔ ↑z ≤ a ∧ a < z + 1 := by |
rw [le_antisymm_iff, le_floor, ← Int.lt_add_one_iff, floor_lt, Int.cast_add, Int.cast_one,
and_comm]
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 177 | 178 | theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by |
simp [Subgraph.spanningCoe]
|
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
noncomputable section
open Polynomial
universe u v w
variable {R : Type*} {F... | Mathlib/Algebra/Polynomial/Splits.lean | 164 | 165 | theorem splits_id_iff_splits {f : K[X]} : (f.map i).Splits (RingHom.id L) ↔ f.Splits i := by |
rw [splits_map_iff, RingHom.id_comp]
|
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 133 | 136 | theorem nonMemberSubfamily_nonMemberSubfamily :
(𝒜.nonMemberSubfamily a).nonMemberSubfamily a = 𝒜.nonMemberSubfamily a := by |
ext
simp
|
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe w w' u u' v v'
variable {R : Type u} {R' : Type u'} {M M₁ : Type v} {M' : Type v'}
open Cardinal Submodule Function... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 185 | 193 | theorem lift_rank_le_of_injective_injective
(i : R' →+* R) (j : S →+* S') (hi : Injective i) (hj : Injective j)
(hc : (j.comp (algebraMap R S)).comp i = algebraMap R' S') :
lift.{v'} (Module.rank R S) ≤ lift.{v} (Module.rank R' S') := by |
refine _root_.lift_rank_le_of_injective_injective i j
(fun _ _ ↦ hi <| by rwa [i.map_zero]) hj fun r _ ↦ ?_
have := congr($hc r)
simp only [RingHom.coe_comp, comp_apply] at this
simp_rw [smul_def, AddMonoidHom.coe_coe, map_mul, this]
|
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 152 | 163 | theorem trans_refl_reparam (p : Path x₀ x₁) :
p.trans (Path.refl x₁) =
p.reparam (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transReflReparamAux_zero) (Subtype.ext transReflReparamAux_one) := by |
ext
unfold transReflReparamAux
simp only [Path.trans_apply, not_le, coe_reparam, Function.comp_apply, one_div, Path.refl_apply]
split_ifs
· rfl
· rfl
· simp
· simp
|
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 421 | 429 | theorem norm_lineDeriv_le_of_lip' {f : E → F} {x₀ : E}
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖ := by |
apply norm_deriv_le_of_lip' (by positivity)
have A : Continuous (fun (t : 𝕜) ↦ x₀ + t • v) := by continuity
have : ∀ᶠ x in 𝓝 (x₀ + (0 : 𝕜) • v), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖ := by simpa using hlip
filter_upwards [(A.continuousAt (x := 0)).preimage_mem_nhds this] with t ht
simp only [preimage_setOf_eq, add_... |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 87 | 89 | theorem iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s := by |
simp_rw [ρ.IicSnd_apply _ hs, Measure.iInf_rat_gt_prod_Iic hs]
|
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 219 | 220 | theorem range_IciExtend (f : Ici a → β) : range (IciExtend f) = range f := by |
simp only [IciExtend, range_comp f, range_projIci, range_id', image_univ]
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.Ring.Ideal
#align_import topology.algebra.uniform_ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open scoped Classica... | Mathlib/Topology/Algebra/UniformRing.lean | 195 | 200 | theorem map_smul_eq_mul_coe (r : R) :
Completion.map (r • ·) = ((algebraMap R A r : Completion A) * ·) := by |
ext x
refine Completion.induction_on x ?_ fun a => ?_
· exact isClosed_eq Completion.continuous_map (continuous_mul_left _)
· simp_rw [map_coe (uniformContinuous_const_smul r) a, Algebra.smul_def, coe_mul]
|
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u → Type... | Mathlib/Data/Option/Defs.lean | 96 | 97 | theorem mem_toList {a : α} {o : Option α} : a ∈ toList o ↔ a ∈ o := by |
cases o <;> simp [toList, eq_comm]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 519 | 525 | theorem setToSimpleFunc_mono_left' (T T' : Set α → E →L[ℝ] G'')
(hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →ₛ E)
(hf : Integrable f μ) : setToSimpleFunc T f ≤ setToSimpleFunc T' f := by |
refine sum_le_sum fun i _ => ?_
by_cases h0 : i = 0
· simp [h0]
· exact hTT' _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hf h0) i
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 1,071 | 1,081 | theorem IsCompact.prod {t : Set Y} (hs : IsCompact s) (ht : IsCompact t) :
IsCompact (s ×ˢ t) := by |
rw [isCompact_iff_ultrafilter_le_nhds'] at hs ht ⊢
intro f hfs
obtain ⟨x : X, sx : x ∈ s, hx : map Prod.fst f.1 ≤ 𝓝 x⟩ :=
hs (f.map Prod.fst) (mem_map.2 <| mem_of_superset hfs fun x => And.left)
obtain ⟨y : Y, ty : y ∈ t, hy : map Prod.snd f.1 ≤ 𝓝 y⟩ :=
ht (f.map Prod.snd) (mem_map.2 <| mem_of_supers... |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 47 | 49 | theorem nhds_eq_update : (𝓝 : Γ₀ → Filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) := by |
rw [nhds_nhdsAdjoint, sup_of_le_right]
exact le_iInf₂ fun γ hγ ↦ le_principal_iff.2 <| zero_lt_iff.2 hγ
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 371 | 420 | theorem hasFTaylorSeriesUpTo_iteratedFDeriv :
HasFTaylorSeriesUpTo ⊤ f (fun v n ↦ f.iteratedFDeriv n v) := by |
classical
constructor
· simp [ContinuousMultilinearMap.iteratedFDeriv]
· rintro n - x
suffices H : curryLeft (f.iteratedFDeriv (Nat.succ n) x) = (∑ e : Fin n ↪ ι,
((iteratedFDerivComponent f e.toEquivRange).linearDeriv
(Pi.compRightL 𝕜 _ Subtype.val x)) ∘L (Pi.compRightL 𝕜 _ Subtype... |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,137 | 1,137 | theorem mk_equiv_of_eq : #(α ≃ β) = 2 ^ #α := by | rw [mk_equiv_of_lift_eq (lift_inj.mpr eq), lift_id]
|
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 46 | 49 | theorem Duplicate.mem (h : x ∈+ l) : x ∈ l := by |
induction' h with l' _ y l' _ hm
· exact mem_cons_self _ _
· exact mem_cons_of_mem _ hm
|
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open scoped MeasureTheory ENNReal NNReal Topology
open MeasureTheory MeasureTheory... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 470 | 471 | theorem dimH_preimage (e : X ≃ᵢ Y) (s : Set Y) : dimH (e ⁻¹' s) = dimH s := by |
rw [← e.image_symm, e.symm.dimH_image]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variab... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 750 | 751 | theorem one_lt_rpow_iff_of_pos (hx : 0 < x) : 1 < x ^ y ↔ 1 < x ∧ 0 < y ∨ x < 1 ∧ y < 0 := by |
rw [rpow_def_of_pos hx, one_lt_exp_iff, mul_pos_iff, log_pos_iff hx, log_neg_iff hx]
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 408 | 445 | theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
(eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
(uif :
∀ f g : Multiset α,
(∀ x ∈ f, Irreducible x) →
(∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.R... |
apply uif
· exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
· exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
calc
Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
rw [hx, Multiset.prod_cons]; exact hfx.2.... |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 116 | 121 | theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) :
↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by |
change _ = ite _ _ _
rw [if_neg, preimage_image_eq, if_pos hs]
· exact Option.some_injective _
· rintro ⟨x, _, ⟨⟩⟩
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 990 | 996 | theorem integral_Iic_sub_Iic (ha : IntegrableOn f (Iic a) μ) (hb : IntegrableOn f (Iic b) μ) :
((∫ x in Iic b, f x ∂μ) - ∫ x in Iic a, f x ∂μ) = ∫ x in a..b, f x ∂μ := by |
wlog hab : a ≤ b generalizing a b
· rw [integral_symm, ← this hb ha (le_of_not_le hab), neg_sub]
rw [sub_eq_iff_eq_add', integral_of_le hab, ← integral_union (Iic_disjoint_Ioc le_rfl),
Iic_union_Ioc_eq_Iic hab]
exacts [measurableSet_Ioc, ha, hb.mono_set fun _ => And.right]
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 327 | 338 | theorem sum_le {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α} :
s.Nonempty → (∀ i ∈ s, padicNorm p (F i) ≤ t) → padicNorm p (∑ i ∈ s, F i) ≤ t := by |
classical
refine s.induction_on (by rintro ⟨-, ⟨⟩⟩) ?_
rintro a S haS IH - ht
by_cases hs : S.Nonempty
· rw [Finset.sum_insert haS]
exact
padicNorm.nonarchimedean.trans
(max_le (ht a (Finset.mem_insert_self a S))
(IH hs fun b hb ↦ ht b (Finset.mem_insert_of_mem hb)... |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 138 | 142 | theorem mul_polyOfInterest_aux2 (n : ℕ) :
(p : 𝕄) ^ n * wittMul p n + wittPolyProdRemainder p n = wittPolyProd p n := by |
convert mul_polyOfInterest_aux1 p n
rw [sum_range_succ, add_comm, Nat.sub_self, pow_zero, pow_one]
rfl
|
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,437 | 1,439 | theorem L1.dist_eq_integral_dist (f g : α →₁[μ] H) : dist f g = ∫ a, dist (f a) (g a) ∂μ := by |
simp only [dist_eq_norm, L1.norm_eq_integral_norm]
exact integral_congr_ae <| (Lp.coeFn_sub _ _).fun_comp norm
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 615 | 623 | theorem rank_pow_quot_aux [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
{i : ℕ} (hi : i < e) :
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
Module.rank (R ⧸ p) (S ⧸ P) +
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ (i + 1))) := by |
letI : Field (R ⧸ p) := Ideal.Quotient.field _
rw [← rank_range_of_injective _ (powQuotSuccInclusion_injective f p P i),
(quotientRangePowQuotSuccInclusionEquiv f p P hP0 hi).symm.rank_eq]
exact (rank_quotient_add_rank (LinearMap.range (powQuotSuccInclusion f p P i))).symm
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 212 | 212 | theorem top_mul' : ∞ * a = if a = 0 then 0 else ∞ := by | convert WithTop.top_mul' a
|
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 549 | 558 | theorem length_think (s : Computation α) [h : Terminates s] : length (think s) = length s + 1 := by |
apply le_antisymm
· exact Nat.find_min' _ (Nat.find_spec ((terminates_def _).1 h))
· have : (Option.isSome ((think s).val (length (think s))) : Prop) :=
Nat.find_spec ((terminates_def _).1 s.think_terminates)
revert this; cases' length (think s) with n <;> intro this
· simp [think, Stream'.cons] at... |
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
#align_import analysis.special_functions.trigonometric.deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open s... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 711 | 711 | theorem sinh_nonpos_iff : sinh x ≤ 0 ↔ x ≤ 0 := by | simpa only [sinh_zero] using @sinh_le_sinh x 0
|
import Mathlib.Init.Order.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.Attr.Register
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Unique
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc95... | Mathlib/Logic/Nontrivial/Basic.lean | 90 | 93 | theorem nontrivial_at (i' : I) [inst : ∀ i, Nonempty (f i)] [Nontrivial (f i')] :
Nontrivial (∀ i : I, f i) := by |
letI := Classical.decEq (∀ i : I, f i)
exact (Function.update_injective (fun i ↦ Classical.choice (inst i)) i').nontrivial
|
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 511 | 514 | theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by |
refine ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => ?_⟩
simp [constFormalMultilinearSeries_apply hn]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 859 | 869 | theorem Integrable.measure_norm_ge_lt_top {f : α → β} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
μ { x | ε ≤ ‖f x‖ } < ∞ := by |
rw [show { x | ε ≤ ‖f x‖ } = { x | ENNReal.ofReal ε ≤ ‖f x‖₊ } by
simp only [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe, ENNReal.coe_le_coe, coe_nnnorm]]
refine (meas_ge_le_mul_pow_snorm μ one_ne_zero ENNReal.one_ne_top hf.1 ?_).trans_lt ?_
· simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hε
appl... |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 | theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by |
refine (em _).imp (dense_of_not_isolated_zero S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε0, hε⟩
exact cyclic_of_isolated_zero ε0 (disjoint_left.2 hε)
|
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.normed_space.hahn_banach.separation from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 84 | 112 | theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
(disj : Disjoint s t) : ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b := by |
obtain rfl | ⟨a₀, ha₀⟩ := s.eq_empty_or_nonempty
· exact ⟨0, 0, by simp, fun b _hb => le_rfl⟩
obtain rfl | ⟨b₀, hb₀⟩ := t.eq_empty_or_nonempty
· exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩
let x₀ := b₀ - a₀
let C := x₀ +ᵥ (s - t)
have : (0 : E) ∈ C :=
⟨a₀ - b₀, sub_mem_sub ha₀ hb₀, by simp_rw [x₀... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 85 | 92 | theorem IsPiSystem.insert_empty {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
IsPiSystem (insert ∅ S) := by |
intro s hs t ht hst
cases' hs with hs hs
· simp [hs]
· cases' ht with ht ht
· simp [ht]
· exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)
|
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 62 | 68 | theorem eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by |
obtain rfl | hm := eq_or_ne m 0
· exact Int.zero_dvd.1 h1
rcases h1 with ⟨d, rfl⟩
apply mul_eq_zero_of_right
rw [← abs_lt_one_iff, ← mul_lt_iff_lt_one_right (abs_pos.mpr hm), ← abs_mul]
exact lt_of_lt_of_le h2 (le_abs_self m)
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 793 | 797 | theorem tendsto_translationNumber (x : ℝ) :
Tendsto (fun n : ℕ => ((f ^ n) x - x) / n) atTop (𝓝 <| τ f) := by |
rw [← translationNumber_conj_eq' (translate <| Multiplicative.ofAdd x)]
refine (tendsto_translation_number₀ _).congr fun n ↦ ?_
simp [sub_eq_neg_add, Units.conj_pow']
|
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 135 | 137 | theorem coe_smul (r : ℝ≥0) : (r : ℝ) • j = r • j := by |
-- Porting note: replaced `show`
rw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,259 | 1,284 | theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
(h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ := by |
simp_rw [← iSup_apply]
let p : α → (β → ENNReal) → Prop := fun x f' => Directed LE.le f'
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by
filter_upwards [] with x i j
obtain ⟨z, hz₁, hz₂⟩ := h_directed i j
exact ⟨z, hz₁ x, hz₂ x⟩
have h_ae_seq_directed : Directed LE.le (aeSeq hf p) := by
intro b₁ b₂... |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 623 | 624 | theorem tendsto_inv_nhdsWithin_Ici_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) := by |
simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ici _ _ _ _ a⁻¹
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,042 | 1,044 | theorem isOpenMap_sum {f : X ⊕ Y → Z} :
IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by |
simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp]
|
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